In telecommunication systems, squaring circuits--i.e., circuits which produce an output which is proportional to the square of an input--are useful in a variety of contexts. Since squaring a sine wave doubles its frequency, a squaring circuit is useful for shifting the frequency of an incoming carrier wave modulated by a signal to a higher frequency range or band. Such shifting is useful, for example, in a modem in which it is desired to process an incoming signal from a transmission channel in which the information is transmitted by phase-shift-keying modulation of the carrier, i.e., by imposing phase shifts upon the carrier wave at regular time intervals (T). The magnitude of each such phase shift can be one of a number of discrete possible amounts, illustratively four, each of these possible amounts of phase shift representing a different possible symbol of information being transmitted. Illustratively, the carrier has a frequency of 1200 Hz, and the phase shifts are imposed every 1/600 second. Thus, illustratively the transmission of information occurs at a symbol frequency or rate of 600 Hz(=1/T).
In order to demodulate the incoming signal and thus to detect the symbols being transmitted by the channel, it is necessary to detect the magnitude of each of the phase shifts on the carrier. In so doing, it is desirable to have a circuit arrangement for squaring the incoming signal in order to produce a signal with frequency components at the symbol rate (1/T) representing the stream of phase shifts; and bandpass filters are then used to remove the carrier wave while preserving the symbol rate information. Illustratively, a bandpass filter centered at the symbol frequency (of 1/T=600 Hz) can be used, whereby the carrier frequency (1200 Hz) is doubled (to 2400 Hz) and is suppressed by the filter, while the symbol rate information (600 Hz) is passed through the filter. Ordinarily, the carrier frequency is higher than the symbol rate. Therefore, since squaring a carrier wave doubles its frequency (but leaves the symbol rate the same), squaring the incoming signal before filtering is thus useful for doubling the carrier frequency (from 1200 to 2400 Hz) and thus separating the carrier further in frequency from the symbol rate (600 Hz). A narrow bandpass filter (i.e., with a quality Q of typically about 10 or more) centered at the symbol rate will then be able to remove more nearly completely the carrier wave component, thus doubled (or more) in frequency, from the symbol rate components, since such a bandpass filter in general more completely removes components whose frequencies are more distant from the center frequency of the pass band. After the original incoming signal (carrier with phase shifts) has thus been squared and filtered, the remaining symbol rate components are relatively free from any carrier and thus are available for further processing.
In the foregoing squaring of the incoming signal, it is important that the squaring be pure and hence not yield any significant amount of first harmonic (fundamental) component (at 1200 Hz); otherwise the desired frequency separation of symbol information from the carrier will not be as complete as desired, and hence the symbol information emanating from the bandpass filter will be undesirably contaminated with the carrier.
To clarify the problem, it is known that a squaring circuit's principle of operation is governed by the trigonometric identity: EQU 2(cos .omega.t).sup.2 =1+cos 2.omega.t (1)
where .omega. is the radial frequency, i.e., EQU .omega.=2.pi.f
where f is the cycle frequency. Thus, squaring a carrier signal of frequency f produces a d.c. component and a second harmonic component (cos 2.omega.t), but no first harmonic. On the other hand, a trigonometric identity such as: EQU 4(cos .omega.t).sup.3 =2 cos .omega.t+cos 3.omega.t (2)
shows that cubing the input (i.e., raising the input to the third power) results in an undesired first harmonic (as well as third harmonic) component in the output. Similarly, any odd integral power component in the output causes the output to contain undesired first harmonic. Thus a desirable squaring circuit does not produce any substantial amount of linear component, or any substantial amount of odd integral power components, in the output relative to the input.
In a paper by O. A. Seriki and R. W. Newcomb entitled "Direct-Coupled MOS Squaring Circuit," published in IEEE Journal of Solid-State Circuits, Vol. SC-14, pp. 766-768 (1979), a squaring circuit is described using MOS (metal oxide semiconductor) transistor technology. That circuit, however, successfully, removes the undesired linear term only if the voltage supplies (V.sub.DD and V.sub.SS) for the transistors in the circuit are carefully controlled and adjusted during operation so that (V.sub.DD +V.sub.SS) is essentially zero. For example, V.sub.DD =+4.0 volts and V.sub.SS =-4.0 volts. But, such control of voltage supplies would necessitate the addition of undesirably complex control circuitry, because in the present state of the art the ordinary voltage source fluctuates by as much as 5 percent during operation, and such voltage fluctuations would produce an undesirable corresponding amount, as much as about 100 percent, first harmonic in the output of that squaring circuit.
Therefore, it would be desirable to have a squaring circuit which substantially removes first harmonic (i.e., the linear component) even in the presence of voltage supply fluctuations of as much as about .+-.10 percent, i.e., well above the amount of ordinary fluctuations. By "substantially removes" it is meant that less than about 5 percent first harmonic amplitude relative to second harmonic is present in the output.